If p ≠ 0 and q ≠ 0 are the roots of the equation x² + px + q = 0, then what is p² + q² equal to?
5
Problem
If p ≠ 0 and q ≠ 0 are the roots of the equation x² + p x + q = 0, find p² + q².
Solution (step-by-step)
- By Vieta's formulas for x² + p x + q = 0, the sum of the roots equals -p and the product equals q.
- But the roots themselves are p and q. Hence
p + q = -p and p q = q
- From p q = q we get q (p - 1) = 0. Since q ≠ 0, it follows that p = 1.
- Substitute p = 1 into p + q = -p:
1 + q = -1 so q = -2
Finally compute
p² + q² = 1² + (-2)² = 1 + 4 = 5
Final answer: p² + q² = 5
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